I have an embarrassing question... math has always interested me but by luck and circumstance I have had a pretty successful career without needing anything beyond college Algebra. Can anyone recommend a curriculum that a busy adult may be able to follow in their spare time? I just want to fill in blanks and explore what is possible.
For me, I set out to write a flight simulator from scratch following a book [1]. I learned a lot of linear algebra that way. Such that when I later took Calculus III in college (it was linear algebra) I aced the class.
(Also, reinforced what I remembered from Trig, I finally fully grokked it. Except for trig identities, ha ha.)
The book was targeting DOS and C++. I wrote it in C for the Macintosh. (That required that I figured out what was going on.)
Go write one in JavaScript using an HTML5 canvas as your buffer. (I made a quick pass at doing something similar [2] — but do it on your own, don't follow my link below or look for the sources for Phosphor3D on GitHub.)
College algebra is just rehashed high school algebra.
Math is a broad subject. This is something LLMs are actually reasonably good at: ask them for textbook recommendations, and get into a dialogue about which sub-areas of math you're interested in and what level you're currently at, whether you want pure math (more theorem-proof focused) or applied math (more practice solving concrete problems, e.g. finding lots of derivatives and integrals). Toss in names of books that have been recommended and ask where they fit in to the LLM's other recommendations.
LLMs don't understand the math, but they're trained on a lot of discussions and recommendations for math books, and have a reasonably good sense of what level different books are at.
Download multiple recommendations in each area and try them all out. Seeing how different authors start out approaching largely the same material will help you conceptualize it better than just relying on a single approach. There's no universal "right" book to learn from. I wouldn't buy non-free textbooks without trying them out first.
Youtube has a lot of math lecture series, which can help if you're stuck on a particular point, but they're not the same as doing problem sets yourself.
LLMs DO understand the math. At least Claude does. Seems to be able to solve linear systems, invert matrices, and do a significant amount calculus, and handle some seriously advanced math problems. I haven't probed the outer limits, but, so far, Claude has handled all the math problems I've given it so far.
Even if a deep thinking LLM like Opus can get some math questions right when that depends on identifying the type of problem and applying a learned procedure, it's not going to be able to evaluate the pedagogy of math books it's never encountered, or at most was fringe material in its training set.
I'm also referring to the faster models, not the slow and expensive deep thinking ones which I have little experience with. I don't see how reasoning would enable deep thinking models to meaningfully evaluate textbook pedagogy, either.
It seems pretty silly to make pronouncements about what LLMs are capable of doing if the sum total of your experience is casual use of the cheapest and least capable LLMs. (ChatGPT: measured IQ of about 70, vs. measured IQs of 120+ for more capable models, some of which are available for free).
They DO understand what they are doing. When I ask it to solve math problems, it goes through the several (many) steps involved (e.g. e.g. "apply the chain rule" while doing partial differentiation on a term in a Jacobian matrix). It gets pretty tedious when solving systems of linear equations, where it goes through each step of the Gauss-Jordan elimination while doing an LU decomposition, row by row. But one learns to ignore the blah-blah. Step by step, in absolutely ridiculous detail. The point: they absolutely 100% understand what they are doing, and understand it in minute detail.
It's clearly NOT regurgitating something that it has literally seen before, because the level of detail is beyond ridiculous for a human. It is applying generalized rules to specific concrete problems, and doing so with some level of strategic thinking.
Where did it learn those generalized principles, and how did it learn to do that? With absolute certainty, there are math textbooks among the materials they have been trained on. And they certainly learned it from SOMEWHERE. Probably math textbooks. How did they learn to generalize and think strategically? Well, that's the big mystery, isn't it? But they do.
The very best models achieve high scores on Math Olympiad problem sets (so competitive with some of the best minds on the planet). And Terrence Tau (greatest living mathematician) declares state-of-the-art models to be "better than most of my post-graduate students".
And what they can and cannot do is increasing by leaps and bounds on a weekly or monthly basis right now. It's hard to keep up. I frequently find that they can do things this week, that they could not do a week or a month ago.
Startling, and quite utterly amazing.
Most of the time, I am using Claude Sonnet 4.5 as my coding agent, for which I pay $10/month. Measured IQ of 110, I think, with an IQ of 120 if you flip it into thinking mode. But only because there isn't enough undergraduate level mathematics in a standard IQ test. Claude Sonnet 4.5 is also available for free here: https://claude.ai/chats (during periods of heavy load, it may fall back to simpler models). I often use the free web interface instead of the Coding Agent interface for math problems, because it's easier to read mathematical equations in the browser version. version). And I generally use the free version of Claude instead of Google Search these days.
You're arguing things I didn't argue. The top-level comment wasn't about how to do some math problem that Sonnet or even Opus is capable of; it was about math book recommendations, and I was specifically mentioning that even though the LLM won't understand the math pedagogy behind why one book might be better than another, it's trained on enough commentary that it will give good recommendations (or anti-recommendations) for any well-known textbooks.
My experience with people who have LLM subscriptions of any kind is that they use them all the time and would immediately ask an LLM that kind of question, rather than asking on a web forum that's not even dedicated to math. So I think it's a fair presumption that someone asking that question doesn't have access to the best commercial models.
On the largely irrelevant question of what math LLMs can do, although Opus may do better, Sonnet can follow procedures sometimes but not consistently. It has blind spots and can't scale procedures; beyond certain numbers or dimensions or problem complexity, it just guesses (wrong). And those limits are quite low. If you want 2 simple examples:
LLMs follow procedures, but whimsically. Better LLMs will be less whimsical, but they still won't be fully competent unless they digest questions into more formal terms and then interface with an engine like Wolfram.
I wanted to brush up on my mostly forgotten high school math and found a free online course which used this ALEKS [1] tool. It was perfect for my situation.
It was some course on edx [2]. Can't find it right now, but you might find another course which uses it.
I like solving math puzzles, and I'm often realizing that I'm missing something I've forgotten since middle school or high school - some formula that I know exists that would make a problem trivial. I'd like to re-learn that stuff, especially as my kiddo starts advancing in math classes.
mathacademy.com is pretty awesome IMO. one hint: don't take notes/allow yourself to refer to notes during quizzes and reviews, the point is to be able to recall using just your brain, even if your progress is much slower that way, you'll learn a lot better.
Many studies concur that the act of taking notes dramatically improves your ability to remember what you wrote, even if you never look at the notes again. You generally don't have time to transcribe a lecture like a stenographer would, so note taking requires you to understand the material enough to summarize it and describe it in your own words.
Don't rely on your notes come quiz time, if you can get away with it, but by all means do take them during the learning process. And in the real world, you're allowed to refer back to your notes as often as you want to.
I've heard of adults going through Khan Academy's math curriculum for fun to refresh their math skills. It goes from K to 12 and up to 2nd or 3rd year university math, including calculus and such.
